Completion of a Banach space with respect to a different norm

Question

Let $(X,|\cdot|_X)$ be a Banach space. Define a space $Y$ as the completion of $X$ under a norm $$|u|_Y = |u|_X + |Tu|_Z$$ where $T:X \to Z$ is a linear continuous map where $X \subset Z$ is a continuous embedding.

What can I say about this new space $Y$? Is it true that: if $v \notin X$, and if the $Y$-norm of $v$ is finite, then $v \in Y?$

Answer 1

Since $T$ is continuous, we have $|Tu|_Z\leq c|u|_X.$ Hence $$|u|_X\leq|u|_Y\leq |u|_X(1+c),$$ i.e. the norms $|\cdot|_X,|\cdot|_Y$ are equivalent. Thus the completion of $X$ w.r.t. $|\cdot|_Y$ is canonically isomorphic to $X.$